The virtual $R$-matrix restriction conjecture

Let $(V,\widehat{V})$ and $(V',\widehat{V}')$ be simple aligned virtual crystals, where $V$ and $V'$ are $X$-crystals and the hatted crystals are $Y$-crystals. Let

$$\widehat{R}:\widehat{V}\otimes\widehat{V}'\to\widehat{V}'\otimes\widehat{V}$$

be the type $Y$ combinatorial $R$-matrix. The virtual $R$-matrix conjecture. The restriction of $\widehat{R}$ defines a map

$$R^v:V\otimes V'\to V'\otimes V.$$

Equivalently, one needs $\widehat{R}(V\otimes V')\subseteq V'\otimes V$. If this holds, uniqueness of the crystal $R$-matrix gives the commuting identification with the $X$-crystal $R$-matrix. The source presents this as a definition-conjecture and gives no resolution status.